Hilbert transforms

The control has been performed by contact, in oblique transverse waves with an angle of 45°. In this test, we have inserted Hilbert transform for detecting envelops. 

The sign alternatives depend on the location of the zeros (or singulai ities) of x i). The above conjugate, or reciprocal, relations aie the main results in this section. When Eqs. (9) and (10) hold, ln x(f) and argx(t) are Hilbert transforms [245,246]. 

Fig. 4.8. Graph of Hilbert transform A(e). Solutions (x) of A(e) = e — eocr locate localized-state energies s. Reprinted from Davison and Huang (1974) with permission from Elsevier.

Marshall, A. G., Ed. "Fourier, Hadamard, and Hilbert Transforms in Chemistry , Plenum PressiNew York, 1982. 

J. A. De Haseth, Fourier Transform Infrared Spectrometry, p 387 in Fourier, Hadamard and Hilbert Transformation in Chemistry (A. G. Marshall, ed.) Plenum, New York, 1982. 

This method is based on the Kramers-Kronig relation (1), saying that both s and e" carry the same information about relaxation processes and are related by a Hilbert transformation ... 

The real and imaginary parts of Ru(z) are therefore Hilbert transforms of each other,... 

High-pass filter, 156, 160 Hilbert transform definition, 409 Householder matrix, 125 Huffman coding, 65 Hybrid filter bank, 56 Hyperparameters, 182... 

Marshall,A.G. In Fourier. Hadamard and Hilbert Transforms in Chemistry Marshall,A.G.,Ed. Plenum Press New York, 1982 Chapter 1. 

Luckily, the real and imaginary parts of the complex dielectric permittivity are not independent of each other and are connected by means of the Kramers-Kronig relations [11]. This is one of the most commonly encountered cases of dispersion relations in linear physical systems. The mathematical technique entering into the Kramers-Kronig relations is the Hilbert transform. Since dc-conductivity enters only the imaginary component of the complex dielectric permittivity the static conductivity can be calculated directly from the data by means of the Hilbert transform. 

The result shows that the dc-conductivity can be computed by using the Hilbert transform applied to the real components of the dielectric permittivity function and subtracting the result from its imaginary components. The main obstacle to the practical application of the Hilbert transform is that the integration in Eq. (48) is performed over infinite limits however, a DS spectroscopy measurement provides values of s (co) only over some finite frequency range. Truncation of the integration in the computation of the Hilbert... 

Practically, the Numerical Hilbert transform can be computed by means of the well-known Fast Fourier Transform (FFT) routine. It is based on the following property of the Hilbert transform [135]. If... 

In other words, Hilbert transformation relations between - /7 c/ 1 (t) and CBA (t) exist, that is,... 

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